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Exterior geometry and function spaces
Specify the obstacle and generalized-ray conventions, Sobolev and time-space notation, Dirichlet realization and spectral scales; connect bounded fractional inputs to exterior localization, traces, compactness and continuation.
1 · Prerequisites
- The Axiom of Choice
- Simultaneous Sobolev extension and localization
- Localized smooth coordinate changes and cutoffs on Fourier Hʳ
- Atlas-independent hypersurface Sobolev norms and H^3/2 tangential derivatives
- Fourier trace on a hyperplane and one-sided H¹ uniqueness
- The sharp trace theorem: boundedness and range in the fractional space
- The trace commutes with smooth cutoffs and is chart local
- Real-order Hˢ as weighted Fourier distributions
- Integer-order W^k,2 and Hᵏ agree with equivalent norms
- Bounded Cᵏ domains and boundary charts
- Finite ambient partitions near compact sets
- Smooth Euclidean hypersurface graphs and compact localization
- The Euclidean tubular neighbourhood theorem
- The normal addition map for a Euclidean submanifold
- Heine-Borel in ℝⁿ: with the Euclidean metric a subset of ℝⁿ is compact if and only if it is closed and bounded, and the proof by bisection uses no choice principle; the same holds on the real line
- Restriction Bessel-potential spaces on polyhedral domains
- Embedded submanifolds and slice charts
- Codimension and hypersurfaces
- Slice-chart restrictions form a smooth atlas
- Test function cutoffs and euclidean localization
- The kernel of the trace is the closure of the test functions
- Zero-boundary Sobolev space as a norm closure
- Compactly supported smooth functions are dense in W^k,p(Rⁿ)
- Zero extension of W₀^1,p has no boundary derivative
2 · Logical flowchart
Arrows show item dependencies. A grouped prerequisite node points to items that use one or more of its members; their exact links are listed above.
3 · Items
Lemma
Canonical higher traces on a bounded boundary or compact interior hypersurface
Lemma
Partial extension across an artificial sphere and annular cutoff gluing
Lemma